Pilkington recorded a version of 'Ain't No Pleasing You' with Chas & Dave in the final episode of An Idiot Abroad, as it was something he wanted to achieve for his 'bucket list'.
The track has yet to receive a release date, but a Facebook page has now gathered thousands of supporters, and the campaign is also active on Twitter and Google+.
The movement wants the single to be granted an official release date on December 18 in order to be eligible to chart for the Christmas week. It will also aim to raise money for The Prostate Cancer Charity.
Ricky Gervais has lent his support to the cause by retweeting the link, but added: "I don't think we're releasing it."
If released, Pilkington will face competition from this year's X Factor winner, the Military Wives Choir, and another campaign aiming to get Nirvana's 'Smells Like Teen Spirit' to the top.
Stephen Merchant recently said that it is unlikely a third series of An Idiot Abroad will be made.
Watch Karl Pilkington and Chas & Dave perform 'Ain't No Pleasing You' below: